Aim How do we handle fractional exponents Do

Aim: How do we handle fractional exponents? Do Now: Fill in the appropriate information 28 = 256 2? 7 = 128 26 = 64 2? 5 = 32 24? = 16 23 = 8 22 = 4 2 21 = 2 -1 = 1/2 ? = 2 -2 = 1/4 2? -3== 1/8

Aim: How do we handle fractional exponents? Do Now: Simplify/Rationalize:

Properties of Exponents Product of Powers Property Power of Power Property Power of Product Property Negative Power Property Zero Power Property Quotients of Powers Property Power of Quotient Property am • an = am+n (am)n = am • n (ab)m = ambm a-n = 1/an, a 0 = 1

Indices, Exponents, and New Power Rules Product of Powers Property am • an = am+n example: 82 • 83 = 82 + 3 = 85 example: x 3 • x 6 = x 3 + 6 = x 9 Power of Product Property (ab)m = am • bm example: (2 • 8)2 = 22 • 82 example: (xy)5 = x 5 • y 5 Power of Quotient Property example:

Types of exponents Positive Integer Exponent an = a • a • • a n factors Zero Exponent 0 Negative Exponent -n Rational Exponent 1/n a 0 = 1 Rational Expo. m/n Negative Rational Exponent beware! - m/n -23/2 is not the same as (-2)3/2

Roots, Radicals & Rational Exponents index radical sign radicand Square Root of a number is one of the two equal factors whose product is that number has an index of 2 can be written exponentially as Every positive real number has two square roots The principal square root of a positive number k is its positive square root, . If k < 0, number is an imaginary

Roots, Radicals & Rational Exponents index radical sign radicand Cube Root of a number is one of the three equal factors whose product is that number has an index of 3 can be written exponentially as k 1/3 principal cube roots

Roots, Radicals & Rational Exponents index radical sign radicand nth Root of a number is one of n equal factors whose product is that number has an index where n is any counting number can be written exponentially as k 1/n principal odd roots principal even roots

Indices and Rational Exponents square root k 1/2 • k 1/2 = k 1/2 + 1/2 = k 1 = k 21/2 • 21/2 = 2 1/2 + 1/2 = 21 = 2 cube root k 1/3 • k 1/3 = k 1/3 + 1/3 = k 1 = k 21/3 • 21/3 = 2 1/3 + 1/3 = 21 = 2 = k 1/n nth root k 1/n • k 1/n. . . = k 1/n + 1/n. . . = k 1 = k n times 81/3 • 81/3 = 8 1/3 + 1/3 = 81 = 8

Fractional Exponents Radicals Fractional Exponents (ab)m = am • bm

Simplifying positive integer exponent rational exponent multiplication law simplify & power law = x 10 - 4 = x 6 division law = x 5 - 2 = x 3

Simplifying – Fractional Exponents A rational expression that contains a fractional exponent in the denominator must also be rationalized. When you simplify an expression, be sure your answer meets all of the given conditions. Conditions for a Simplified Expression 1. It has no negative exponents. 2. It has no fractional exponents in the denominator. 3. It is not a complex fraction. 4. The index of any remaining radical is as small as possible.

Model Problems Rewrite using radicals: Rewrite using rational exponents: Evaluate:

Evaluating Evaluate a 0 + a 1/3 + a -2 when a = 8 80 + 81/3 + 8 -2 replace a with 8 1 + 81/3 + 8 -2 x 0 = 1 1 + 2 + 8 -2 1 + 2 + 1/64 3 1/64 x 1/3 = x–n = 1/xn 8– 2 = 1/82 = 1/64 combine like terms If m = 8, find the value of (8 m 0)2/3 (8 • 80)2/3 (8 • 1)2/3 (8)2/3 =4 replace m with 8 x 0 = 1

Simplifying – Fractional Exponents